We study the non-uniformity of probability measures on the interval and circle. On the interval, we identify the Wasserstein-p distance with the classical \(L^p\)-discrepancy. We thereby derive sharp estimates in Wasserstein distances for the irregularity of distribution of sequences on the interval and circle. Furthermore, we prove an \(L^p\)-adapted Erdős–Turán inequality, and use it to extend a well-known bound of Pólya and Vinogradov on the equidistribution of quadratic residues in finite fields.

中文翻译：

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Wasserstein距离的不规则分布

我们研究了区间和圆上的概率测度的非均匀性。在该间隔上，我们用经典\（L ^ p \）-差异来确定Wasserstein- p距离。因此，对于间隔和圆上序列的不规则分布，我们在Wasserstein距离中得出了清晰的估计。此外，我们证明了\（L ^ p \）自适应的Erdős-Turán不等式，并用它扩展了Pólya和Vinogradov在有限域中二次残基的等分分布上的一个已知界。